Quantum information theory reveals a clear distinction between local and nonlocal correlations through the entanglement across spatially separated subsystems. The orthogonal complement of an unextendible biseparable basis (UBB) consists entirely of genuine multipartite entangled states, representing the most robust form of such nonlocal correlations. In this letter, we provide a sufficient condition for any subspace to be genuinely entangled, enabling the systematic construction of high-dimensional genuinely entangled subspaces (GESs) from UBBs. Our construction yields the largest known GES ever obtained from a UBB. In fact, every state in this subspace is 1-distillable across every bipartition which is one of the crucial result we obtained. Furthermore, we prove that every UBB is indistinguishable under LOCC protocols, underscoring a distinct manifestation of quantum nonlocality. The UBBs we construct exhibit strong nonlocality in this scenario, making cryptographic protocols secure not only against LOCC-based attacks but also against coordinated group attacks. We introduce a no-go condition that certifies such an extreme form of nonlocality. All previously known UBBs satisfy this condition, which may lead to the misconception that strong nonlocality is an inherent property of every UBB. However, we construct a UBB that violates the no-go condition and exhibits locality across certain bipartitions, challenging conventional notions of unextendibility and nonlocality in multipartite quantum systems.Quantum information theory reveals a clear distinction between local and nonlocal correlations through the entanglement across spatially separated subsystems. The orthogonal complement of an unextendible biseparable basis (UBB) consists entirely of genuine multipartite entangled states, representing the most robust form of such nonlocal correlations. In this letter, we provide a sufficient condition for any subspace to be genuinely entangled, enabling the systematic construction of high-dimensional genuinely entangled subspaces (GESs) from UBBs. Our construction yields the largest known GES ever obtained from a UBB. In fact, every state in this subspace is 1-distillable across every bipartition which is one of the crucial result we obtained. Furthermore, we prove that every UBB is indistinguishable under LOCC protocols, underscoring a distinct manifestation of quantum nonlocality. The UBBs we construct exhibit strong nonlocality in this scenario, making cryptographic protocols secure not only against LOCC-based attacks but also against coordinated group attacks. We introduce a no-go condition that certifies such an extreme form of nonlocality. All previously known UBBs satisfy this condition, which may lead to the misconception that strong nonlocality is an inherent property of every UBB. However, we construct a UBB that violates the no-go condition and exhibits locality across certain bipartitions, challenging conventional notions of unextendibility and nonlocality in multipartite quantum systems.

arXiv:2508.17154Physical Review AApr 7, 20261 citation

Multipartite entanglement

Ingemar Bengtsson, Karol Życzkowski

Cambridge University Press eBooksAug 14, 201712 citations

Entangled multipartite systems

Springer eBooksJan 1, 20070 citations

Minimum Cardinalities of Multipartite Unextendible Product Bases

Chenhao Wang

In quantum information theory, the state space of a multipartite quantum system is modeled by a tensor product. In the tensor-product space $\C^{d_1}\otimes\cdots\otimes\C^{d_p}$, a nonzero vector is a \emph{product state} if it can be written as $\ket{\varphi_1}\otimes\cdots\otimes\ket{\varphi_p}$ with $\ket{\varphi_j}\in\C^{d_j}\setminus\{0\}$. An \emph{unextendible product basis} (UPB) is a finite family of pairwise orthogonal product states such that no nonzero product state is orthogonal to all of them. UPBs play a key role in investigating quantum entanglement and nonlocal phenomena. Finding a smallest UPB is a natural extremal problem: it asks how few pairwise orthogonal product states suffice to prevent any further product state from being added. The general minimum-size problem for UPBs has been studied for over two decades since the seminal work of Alon and Lovász. Let $f_m(d_1,\ldots,d_p)$ be the minimum cardinality of a UPB in the space, and let $f_{LB}(d_1,\ldots,d_p)=1+\sum_{j=1}^{p}(d_j-1)$ be the natural lower bound. Alon and Lovász determined exactly when $f_m$ attains the lower bound $f_{LB}$ , but the obstructed multipartite cases remained open in general. We prove a stabilization theorem: for every non-all-qubit system with $p\ge3$ , whenever the natural lower bound $f_{LB}$ is unattainable, the true minimum is exactly $f_{LB}+1$ . Precisely, if the number of even local dimensions $d_i$ is positive and even, and at least one local dimension is greater than two, then $f_m(d_1,\ldots,d_p)=f_{LB}(d_1,\ldots,d_p)+1$ . The proof is built on a unified graph-theoretic framework. Our result, together with earlier work, settles the minimum size problem for UPBs in all finite quantum systems.In quantum information theory, the state space of a multipartite quantum system is modeled by a tensor product. In the tensor-product space $\C^{d_1}\otimes\cdots\otimes\C^{d_p}$, a nonzero vector is a \emph{product state} if it can be written as $\ket{\varphi_1}\otimes\cdots\otimes\ket{\varphi_p}$ with $\ket{\varphi_j}\in\C^{d_j}\setminus\{0\}$. An \emph{unextendible product basis} (UPB) is a finite family of pairwise orthogonal product states such that no nonzero product state is orthogonal to all of them. UPBs play a key role in investigating quantum entanglement and nonlocal phenomena. Finding a smallest UPB is a natural extremal problem: it asks how few pairwise orthogonal product states suffice to prevent any further product state from being added. The general minimum-size problem for UPBs has been studied for over two decades since the seminal work of Alon and Lovász. Let $f_m(d_1,\ldots,d_p)$ be the minimum cardinality of a UPB in the space, and let $f_{LB}(d_1,\ldots,d_p)=1+\sum_{j=1}^{p}(d_j-1)$ be the natural lower bound. Alon and Lovász determined exactly when $f_m$ attains the lower bound $f_{LB}$, but the obstructed multipartite cases remained open in general. We prove a stabilization theorem: for every non-all-qubit system with $p\ge3$, whenever the natural lower bound $f_{LB}$ is unattainable, the true minimum is exactly $f_{LB}+1$. Precisely, if the number of even local dimensions $d_i$ is positive and even, and at least one local dimension is greater than two, then $f_m(d_1,\ldots,d_p)=f_{LB}(d_1,\ldots,d_p)+1$. The proof is built on a unified graph-theoretic framework. Our result, together with earlier work, settles the minimum size problem for UPBs in all finite quantum systems.

quant-pharXiv:2609.05657Sep 8, 20260 citations

\ket{\varphi_1}\otimes\cdots\otimes\ket{\varphi_p}

with $\ket{\varphi_j}\in\C^{d_j}\setminus\{0\}$. An \emph{unextendible product basis} (UPB) is a finite family of pairwise orthogonal product states such that no nonzero product state is orthogonal to all of them. UPBs play a key role in investigating quantum entanglement and nonlocal phenomena. Finding a smallest UPB is a natural extremal problem: it asks how few pairwise orthogonal product states suffice to prevent any further product state from being added. The general minimum-size problem for UPBs has been studied for over two decades since the seminal work of Alon and Lovász. Let

f_m(d_1,\ldots,d_p)

be the minimum cardinality of a UPB in the space, and let

f_{LB}(d_1,\ldots,d_p)=1+\sum_{j=1}^{p}(d_j-1)

be the natural lower bound. Alon and Lovász determined exactly when

f_m

attains the lower bound

f_{LB}

, but the obstructed multipartite cases remained open in general. We prove a stabilization theorem: for every non-all-qubit system with

p\ge3

, whenever the natural lower bound

f_{LB}

is unattainable, the true minimum is exactly

f_{LB}+1

. Precisely, if the number of even local dimensions

d_i

is positive and even, and at least one local dimension is greater than two, then

f_m(d_1,\ldots,d_p)=f_{LB}(d_1,\ldots,d_p)+1

. The proof is built on a unified graph-theoretic framework. Our result, together with earlier work, settles the minimum size problem for UPBs in all finite quantum systems.